Combinatorial trees in Priestley spaces

Richard N. Ball; Aleš Pultr; Jiří Sichler

Commentationes Mathematicae Universitatis Carolinae (2005)

  • Volume: 46, Issue: 2, page 217-234
  • ISSN: 0010-2628

Abstract

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We show that prohibiting a combinatorial tree in the Priestley duals determines an axiomatizable class of distributive lattices. On the other hand, prohibiting n -crowns with n ≥ 3 does not. Given what is known about the diamond, this is another strong indication that this fact characterizes combinatorial trees. We also discuss varieties of 2-Heyting algebras in this context.

How to cite

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Ball, Richard N., Pultr, Aleš, and Sichler, Jiří. "Combinatorial trees in Priestley spaces." Commentationes Mathematicae Universitatis Carolinae 46.2 (2005): 217-234. <http://eudml.org/doc/249533>.

@article{Ball2005,
abstract = {We show that prohibiting a combinatorial tree in the Priestley duals determines an axiomatizable class of distributive lattices. On the other hand, prohibiting $n$-crowns with $n\ge 3$ does not. Given what is known about the diamond, this is another strong indication that this fact characterizes combinatorial trees. We also discuss varieties of 2-Heyting algebras in this context.},
author = {Ball, Richard N., Pultr, Aleš, Sichler, Jiří},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {distributive lattice; Priestley duality; poset; first-order definable; distributive lattice; Priestley duality; poset},
language = {eng},
number = {2},
pages = {217-234},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Combinatorial trees in Priestley spaces},
url = {http://eudml.org/doc/249533},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Ball, Richard N.
AU - Pultr, Aleš
AU - Sichler, Jiří
TI - Combinatorial trees in Priestley spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 2
SP - 217
EP - 234
AB - We show that prohibiting a combinatorial tree in the Priestley duals determines an axiomatizable class of distributive lattices. On the other hand, prohibiting $n$-crowns with $n\ge 3$ does not. Given what is known about the diamond, this is another strong indication that this fact characterizes combinatorial trees. We also discuss varieties of 2-Heyting algebras in this context.
LA - eng
KW - distributive lattice; Priestley duality; poset; first-order definable; distributive lattice; Priestley duality; poset
UR - http://eudml.org/doc/249533
ER -

References

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  1. Adams M.E., Beazer R., Congruence properties of distributive double p -algebras, Czechoslovak Math. J. 41 (1991), 395-404. (1991) Zbl0758.06008MR1117792
  2. Adámek J., Herrlich H., Strecker G., Abstract and concrete categories, Wiley Interscience, New York, 1990. MR1051419
  3. Ball R.N., Pultr A., Forbidden Forests in Priestley Spaces, Cahiers Topologie Géom. Différentielle Catég. 45 1 (2004), 2-22. (2004) Zbl1062.06020MR2040660
  4. Ball R.N., Pultr A., Sichler J., Priestley configurations and Heyting varieties, submitted for publication. Zbl1165.06003
  5. Ball R.N., Pultr A., Sichler J., Configurations in coproducts of Priestley spaces, to appear in Appl. Categ. Structures. Zbl1086.06012MR2141593
  6. Burris S., Sankappanavar H.P., A Course in Universal Algebra, Graduate Texts in Mathematics 78, Springer, New York-Heidelberg-Berlin, 1981. Zbl0478.08001MR0648287
  7. Davey B.A., Priestley H.A., Introduction to Lattices and Order, second edition, Cambridge University Press, New York, 2001. Zbl1002.06001MR1902334
  8. Koubek V., Sichler J., On Priestley duals of products, Cahiers Topologie Géom. Différentielle Catég. 32 (1991), 243-256. (1991) Zbl0774.06006MR1158110
  9. Łoś J., Quelques remarques, théorèmes et problèmes sur les classes définisables d'algèbres, Mathematical interpretation of formal systems, North-Holland, Amsterdam, 1955, pp.98-113. 
  10. Monteiro A., L'arithmetique des filtres et les espaces topologiques, I, II, Notas de Lógica Matemática (1974), 29-30. (1974) Zbl0318.06019
  11. Priestley H.A., Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc. 2 (1970), 186-190. (1970) Zbl0201.01802MR0265242
  12. Priestley H.A., Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc. 324 (1972), 507-530. (1972) Zbl0323.06011MR0300949

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